APPENDIX E
MODELING OF STEADY-STATE TIME-CONCENTRATION
PROFILES
Cory Langston*, DVM, PhD, DACVCP
*College of
Veterinary Medicine
Box 6100
Miss. State, MS
39762-6100
662-325-3432
langston@cvm.msstate.edu
Introduction
The Veterinary Antimicrobial Decision Support (VADS)
System is a collaborative effort to use existing pharmacokinetic,
pharmacodynamic, clinical trial, and pathogen susceptibility information to
create optimum antimicrobial dosage regimens, label and extra-label, for use in
food animals. Generation of drug time-concentration
profiles at steady-state is an integral part of the VADS method of developing
dosage regimens. Because of the need to
make population predictions only datasets that contain estimates of variability
are appropriate for modeling.
A
meta-analysis of population models would be the ideal approach for this
endeavor. A review of the literature
however failed to show any population-pharmacokinetic studies of antimicrobials
in food animals. This finding was not
unexpected. What was surprising however
was the paucity of adequate conventional compartmental models. When they are encountered such models are
used in VADS utilizing the commercial software program Trial Simulator (TS) by
Pharsightã which adds parameter variability into its
estimates. It rapidly became apparent
that by far the most frequently available type of data was single-dose mean +
standard deviation time-concentration tables.
Though modeling of the mean profile was considered, it is fraught with
pitfalls and, more importantly, there is no way to extrapolate and assign the
concentration variability to the variability of the compartmental
parameters. For these reasons, the
superpositioning approach is utilized for most VADS datasets to predict
time-concentration profiles with their associated standard deviations at
steady-state. A Visual Basic computer
program, Super.exe, was written to automate the superpositioning process. The results
are then entered into a Monte Carlo simulation
program to make population predictions.
.
Superpositioning
Basic concepts
and limitations of superpositioning
Superpositioning is a simple yet powerful kinetic
method for predicting a multiple dose time-concentration drug profile from
single-dose data with the intent to predict steady-state concentrations
associated with a specific dose and dosage interval. It assumes that early doses of a drug do not
affect the pharmacokinetics of subsequent doses; that is, time-concentration
points are independent of each other.[i] Fundamentally, to use superpositioning one
simply overlays each dose at the appropriate interval and then sums the
concentrations at each time point. . An example of how this works is seen in the
following table where a total of six doses are given at four hour intervals.
Example 1
The following graph shows the time-concentration profile in
the above example where “Time” is the x-axis and “Total” (cumulative drug
concentration) is plotted on the y-axis.
It must be remembered that in order for
superpositioning to work certain requirements must be met. First, both absorption and elimination must
be first-order processes. Second, one
must be able to account for drug concentrations remaining from prior
doses. It is the latter assumption that
causes most difficulties as, unlike in textbook examples, rarely will one find
a dataset where the drug is measured at regular intervals over an extended
period of time. More commonly, samples
are collected at frequent intervals early in a dosing to better delineate
absorptive or distributive processes and much less frequently in the terminal
phase. In Example 2 below,
consider the single-dose data seen in the table. Note how many missing data points, seen in
red, must be predicted to provide residual dose one concentrations for
subsequent addition to dose two concentrations.
Example 2
Predicting missing time-concentration
data points
Proximal points
When time-concentration points are missing in the
superpositioning process they must be predicted. For concentration data this is done using
linear regression. The default method to
derive the needed point in absorptive or distribution phases is to regress one
existing (single-dose) time-concentration point on either side of the needed
time, henceforth referred to as two-point regression. Note that only real (not previously predicted)
data points are used to create a regression equation. The underlying rationale for the above method
is that the said missing point should lie between existing points on either
side. Though curve stripping and
associated compartmental fitting was considered as an alternative to derive
these missing points, it was felt that such efforts would have minimal benefit
as only Cmax, the concentration at the mid-point of the dosing interval (C50),
and Cmin are of interest in VADS and not absorptive or distributive rate
constant comparisons. This method of regressing
between two points is thus the default approach for predicting missing points in
the proximal portion of the curve.
Time-zero issues
When a point needs to be predicted between the first
measured data point and time-zero (at concentration zero) we encounter an
additional problem for non-intravenous data because one cannot take the logarithm
of zero to represent one of the two points in the regression. Two approaches were considered to deal with
this problem: 1) perform a zero-order
regression of the zero concentration and the first true time point or 2) have the
program decrease the first true concentration 1000 fold to represent zero and
continue with the first-order regression.
Neither approach is ideal, however since first-order processes are a
basic assumption of superpositioning the latter approach is used.
For an intravenous dataset a time-zero concentration must
be determined. While curve-stripping and
regression to produce a mathematical C0 (intercept) was considered,
because concentration-dependent pharmacodynamic targets are based on measured
concentrations it was felt that use of the y-intercept would produce artificially
high Cmax values. As such, for intravenous
datasets time-zero is manually entered as a duplicate of the first nonzero
concentration (and associated standard deviation).
Terminal points
For the terminal portion of the single-dose curve
missing data points are predicted based on standard curve stripping techniques utilizing
a residual sum of squares approach with subsequent first-order regression of
the terminal points. This
“Auto-calculate” feature of the software uses the following steps to accomplish
terminal phase selection:
·
The regression portion of the program first
removes all points proximal to and inclusive of Cmax.
·
It then creates a residual sum of squares for
all possible variations of data allocation into one- and two-compartment
models.
·
An Aikakie’s Information Criteria score is used
to determine the best model which is then output and a graph displayed.
In selected rare cases, if the user disagrees with the
terminal phase selected by this method he may instead specify the data points
from which to create the regression equation.
How far to predict terminal
concentrations
Central to the concept of superpositioning is that one
must account for the drug remaining from prior doses. If this is not done then no accumulation will
occur. How far beyond the dosing
interval such residual concentrations must be estimated is a matter of user
preference. VADS has chosen to use the
more conservative (greater duration of prediction) of either of the following three
techniques:
- Extrapolate
to 2X interval: Extrapolation
to at least twice the desired dosing interval.
- Extrapolate
to specified concentration cut-off:
Extrapolation to at least 25% of the lowest tested NCCLS MIC value
for that antimicrobial.
Concentrations less than that amount are disregarded.
- Assure
curve extends to last measured point:
On rare occasion one encounters a dataset where sampling was
conducted for an extended period of time.
Most commonly these are drug residue studies. When such times extend beyond options 1
and 2, this method will be chosen.
Superpositioning of variability
If concentrations are independent variables then it
follows that their standard deviations (SDs) are independent. So if the SDs at the various times are
uncorrelated, the variance of the sum would equal the sum of the variances,
i.e., variances are additive.
Specifically, in the overlay of SDs each SD is squared, all are summed
across that time point, and the square root of the total provides the estimated
SD at that time.
Predicting missing SD
data points
Wherever there is a missing time-concentration point
there will obviously be a missing standard deviation (SD) point. As mentioned above, for intravenous datasets
just as time-zero is manually entered as a duplicate of the first nonzero
concentration, the SD associated with that point is used for the time-zero SD. Missing proximal points other than time-zero are
predicted using the same method as that of missing concentration data, i.e.,
the two-point regression method.
For missing terminal SDs the approach is to use the coefficient
of variation (CV) of an existing SD.
Though the CV of the last single-dose SD is the default value offered to
the user, he may instead opt for a CV of his choosing. Note that this method of estimating the SD
applies only for missing data points from the last raw data SD to the last
needed point. In those datasets where
the SD of the last measured concentration is zero (usually because only one
animal had a detectable concentration at that time) then the data point with
the last nonzero concentration having a nonzero SD is used to calculate a CV,
which is then used to produce the SD of any remaining terminal points. Missing SDs proximal to the last raw data SD
are made using the default two-point regression method. In non-intravenous datasets where a point
between time-zero and the first true data point must be predicted then the time-zero
SD is assumed to be 1/1000th of the first true data point SD.
Dosage issues
In producing a steady-state plasma time-concentration
profile using superpositioning one may choose to use a dose different from that
used in the single-dose study. To do so,
one simply multiplies the single-dose concentrations by the ratio of the
proposed dose to that used in the single-dose study. The original dose that produced the
single-dose profile is however used in the VADS model. A dosing interval is chosen based on what is
likely to be employed in food animal practice, but typically 12 and 24 hour
intervals are used for most formulations while 48 and 72 hour intervals are use
for more repository formulations. To
assure that steady-state has been reached, the number of doses simulated is
chosen to assure the total simulation time equals or exceeds 7 terminal
half-lives.
Parameters reported
·
The terminal half-life (lz), the data points used in the linear regression to determine
that terminal half-life, and the R2 value and p-value for that
regression.
·
Area under the time-concentration curve for the
single-dose data from time zero to infinity AUC(0-¥).
·
The ratio of the single-dose AUC(0-t)
to AUC(0-¥)
·
The dose used in the single-dose study and
therefore used in all superpositionings.
·
The dosing interval used in the superpositioning
·
The number of doses modeled. The default is the number of doses required
to reach 99% (7 terminal half-lives) of steady-state. Where this default is not followed (e.g.,
only two doses of a repository preparation given) the exception in noted.
·
The method chosen to predict terminal missing
data points. See the above section “How
far to predict terminal concentrations”
·
The coefficient of variation used to extrapolate
the SD of predicted terminal data points that extend beyond the last true data
point. The default value is the CV of
the last true data point.
·
Single-dose trough concentration
·
SD of the single-dose trough concentration
·
Steady-state trough concentration
·
SD of the steady-state trough concentration
·
Steady-state concentration at the midpoint of
that dosing interval (C50)
·
SD of the steady-state C50 concentration
·
Single-dose Cmax (observed) and its associated
Tmax
·
SD of the single-dose Cmax
·
Cmax at steady-state
·
SD of Cmax at steady-state
·
Priming dose ratio based on trough: The steady-state trough divided by the
single-dose trough.
·
Priming dose ratio based on Cmax: The steady-state Cmax divided by the
single-dose Cmax. (Note: only one priming dose ratio is reported on
the web page. The one chosen is based on
the most likely pharmacodynamics of that drug.)
Software validation
The following datasets analyzed by Super.exe were also
analyzed by manual methods to verify the consistency of the results. This is repeated whenever a change in the
Super.exe program is made.
·
Concentration data from examples in Gibaldi
and Shargel & Yu.
·
Procaine penicillin G concentration and SD
datasets
261
260
259
74
290
1865
·
Ampicillin trihydrate concentration and SD
datasets
252
263
264
